# Definition:Cauchy Principal Value/Complex Integral

Jump to navigation
Jump to search

## Definition

Let $f: \R \to \C$ be a bounded complex function.

Then the **Cauchy principal value of $\ds \int f$** is defined as:

- $\PV_{-\infty}^{+\infty} \map f t \rd t := \lim_{R \mathop \to +\infty} \int_{-R}^R \map f t \rd t$

where $\ds \int_{-R}^R \map f t \rd t$ is a complex Riemann integral.

## Also denoted as

Variants of the notation $\PV$ for the **Cauchy principal value** can often be seen, such as:

- $\operatorname {P.V.} \ds \int$

- $\operatorname {p.v.} \ds \int$

- $PV \ds \int$

and so on.

## Source of Name

This entry was named for Augustin Louis Cauchy.

## Technical Note

The $\LaTeX$ code for \(\PV\) is `\PV`

.

## Sources

This page may be the result of a refactoring operation.As such, the following source works, along with any process flow, will need to be reviewed. When this has been completed, the citation of that source work (if it is appropriate that it stay on this page) is to be placed above this message, into the usual chronological ordering.If you have access to any of these works, then you are invited to review this list, and make any necessary corrections.To discuss this page in more detail, feel free to use the talk page.When this work has been completed, you may remove this instance of `{{SourceReview}}` from the code. |

- 2004: James Ward Brown and Ruel V. Churchill:
*Complex Variables and Applications*(7th ed.): $\S 7$